Respuesta :
The question is missing some parts. Here is the complete question.
An ideal gas is contained in a vessel at 300K. The temperature of the gas is then increased to 900K.
(i) By what factor does the average kinetic energy of the molecules change, (a) a factor of 9, (b) a factor of 3, (c) a factor of [tex]\sqrt{3}[/tex], (d) a factor of 1, or (e) a factor of [tex]\frac{1}{3}[/tex]?
Using the same choices in part (i), by what factor does each of the following change: (ii) the rms molecular speed of the molecules, (iii) the average momentum change that one molecule undergoes in a colision with one particular wall, (iv) the rate of collisions of molecules with walls, and (v) the pressure of the gas.
Answer: (i) (b) a factor of 3;
(ii) (c) a factor of [tex]\sqrt{3}[/tex];
(iii) (c) a factor of [tex]\sqrt{3}[/tex];
(iv) (c) a factor of [tex]\sqrt{3}[/tex];
(v) (e) a factor of 3;
Explanation: (i) Kinetic energy for ideal gas is calculated as:
[tex]KE=\frac{3}{2}nRT[/tex]
where
n is mols
R is constant of gas
T is temperature in Kelvin
As you can see, kinetic energy and temperature are directly proportional: when tem perature increases, so does energy.
So, as temperature of an ideal gas increased 3 times, kinetic energy will increase 3 times.
For temperature and energy, the factor of change is 3.
(ii) Rms is root mean square velocity and is defined as
[tex]V_{rms}=\sqrt{\frac{3k_{B}T}{m} }[/tex]
Calculating velocity for each temperature:
For 300K:
[tex]V_{rms1}=\sqrt{\frac{3k_{B}300}{m} }[/tex]
[tex]V_{rms1}=30\sqrt{\frac{k_{B}}{m} }[/tex]
For 900K:
[tex]V_{rms2}=\sqrt{\frac{3k_{B}900}{m} }[/tex]
[tex]V_{rms2}=30\sqrt{3}\sqrt{\frac{k_{B}}{m} }[/tex]
Comparing both veolcities:
[tex]\frac{V_{rms2}}{V_{rms1}}= (30\sqrt{3}\sqrt{\frac{k_{B}}{m} }) .\frac{1}{30} \sqrt{\frac{m}{k_{B}} }[/tex]
[tex]\frac{V_{rms2}}{V_{rms1}}=\sqrt{3}[/tex]
For rms, factor of change is [tex]\sqrt{3}[/tex]
(iii) Average momentum change of molecule depends upon velocity:
q = m.v
Since velocity has a factor of [tex]\sqrt{3}[/tex] and velocity and momentum are proportional, average momentum change increase by a factor of
(iv) Collisions increase with increase in velocity, which increases with increase of temperature. So, rate of collisions also increase by a factor of [tex]\sqrt{3}[/tex].
(v) According to the Pressure-Temperature Law, also known as Gay-Lussac's Law, when the volume of an ideal gas is kept constant, pressure and temperature are directly proportional. So, when temperature increases by a factor of 3, Pressure also increases by a factor of 3.
